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Python数据结构树

Ding Jiaxiong 人气:0

1.示例

树的一些属性:

2.术语及定义

1.定义一:树由节点及连接节点的边构成。

树的属性:

2.定义二:一棵树要么为空,要么由一个根节点和零棵或多棵子树构成,子树本身也是一棵树。

每棵子树的根节点通过一条边连到父树的根节点。

3.实现

3.1 列表之列表

在这里插入图片描述

树的根节点是myTree[0],左子树是myTree[1],右子树是myTree[2]。

# 列表函数
def BinaryTree(r):
    return [r,[],[]] # 根节点r,和两个作为子节点的空列表
# 插入左子树
def insertLeft(root,newBranch):
    t = root.pop(1)
    if len(t) > 1:
        root.insert(1,[newBranch,t,[]])
    else:
        root.insert(1,[newBranch,[],[]])
    return root

## 插入右子树
def insertRight(root , newBranch):
    t = root.pop(2)
    if len(t) > 1:
        root.insert(2,[newBranch,[],t])
    else:
        root.insert(2,[newBranch,[],[]])
    return root
### 树的访问函数
def getRootVal(root):
    return root[0]
def setRootVal(root,newVal):
    root[0] = newVal
def getLeftChild(root):
    return root[1]
def getRightChild(root):
    return root[2]
r = BinaryTree(3)
insertLeft(r,4)
print(r)

3.2节点与引用

定义一个类,其中有根节点和左右子树的属性。

class BinaryTree:
    def __init__(self,rootObj):
        self.key = rootObj
        self.leftChild = None
        self.rightChild = None

    ## 插入左子节点
    def insertLeft(self,newNode):
        if self.leftChild == None:
            self.leftChild = BinaryTree(newNode)
        else:
            t = BinaryTree(newNode)
            t.left = self.leftChild
            self.leftChild = t
    ## 插入右子节点
    def insertRight(self,newNode):
        if self.rightChild == None:
            self.rightChild = BinaryTree(newNode)
        else:
            t = BinaryTree(newNode)
            t.right = self.rightChild
            self.rightChild = t
    ## 访问函数
    def getRightChild(self):
        return self.rightChild

    def getLeftChild(self):
        return self.leftChild

    def setRootVal(self,obj):
        self.key = obj
    def getRootVal(self):
        return self.key

4.二叉树的应用

4.1解析树

解析树构建器:

import operator

from pythonds.basic import Stack
from pythonds.trees import BinaryTree
def buildParseTree(fpexp):
    fplist = fpexp.split()
    pStack = Stack()
    eTree = BinaryTree("")
    pStack.push(eTree)
    currentTree = eTree

    for i in fplist:
        if i == "(":
            currentTree.insertLeft("")
            pStack.push(currentTree)
            currentTree = currentTree.getLeftChild()

        elif i not in "+-*/)":
            currentTree.setRootVal(eval(i))
            parent = pStack.pop()
            currentTree = parent
        elif i in "+-*/":
            currentTree.setRootVal(i)
            currentTree.insertRight("")
            currentTree = currentTree.getRightChild()
        elif i == ")":
            currentTree = pStack.pop()
        else:
            raise ValueError("Unkown Operator :" + i )
    return eTree

## 计算二叉解析树的递归函数
def evaluate(parseTree):
    opers = {
        "+":operator.add,"-":operator.sub,
        "*":operator.mul,"/":operator.truediv
    }
    
    leftC = parseTree.getLeftChild()
    rightC = parseTree.getRightChild()
    
    if leftC and rightC:
        fn = opers[parseTree.getRootVal()]
        return fn(evaluate(leftC),evaluate(rightC))
    else:
        return parseTree.getRootVal()

4.2树的遍历

前序遍历算法实现为外部函数:

def preorder(tree):
    if tree:
        print(tree.getRootVal())
        preorder(tree.getLeftChild())
        preorder(tree.getRightChild)

前序遍历算法实现为BinaryTree类的方法

def preorder(self):
    print(self.key)
    if self.leftChild:
        self.leftChild.preorder()
    if self.rightChild:
        self.rightChild.preorder()

后序遍历函数

def postorder(tree):
    if tree != None:
        postorder(tree.getLeftChild())
        postorder(tree.getRightChild())
        print(tree.getRootVal())

中序遍历函数

def inorder(tree):
    if tree != None:
        inorder(tree.getLeftChild())
        print(tree.getRootVal())
        inorder(tree.getRightChild())

5.利用二叉堆实现优先级队列

队列一个重要的变体 → 优先级队列。和队列一样,优先级队列从头部移除元素,不过元素的逻辑顺序是由优先级决定的,优先级最高的元素在最前,最低的元素在最后。

实现优先级队列的经典方法 → 二叉堆。入队和出队操作均可达到O(logn)

结构属性:

堆的有序性:对于堆中任意元素x及其父元素p,p都不大于x。

堆操作

代码实现:

class EchaDui:
    # 新建二叉堆
    def __init__(self):
        self.heapList = [0]
        self.currentSize = 0
    
    def percUp(self,i):
        while i // 2 > 0:
            if self.heapList[i] < self.heapList[i // 2]:
                tmp = self.heapList[i // 2]
                self.heapList[i // 2] = self.heapList[i]
                self.heapList[i] = tmp
                
            i = i // 2
    # 新加元素
    def insert(self,k):
        self.heapList.append(k)
        self.currentSize = self.currentSize + 1
        self.percUp(self.currentSize)
        
    def percDown(self,i):
        while (i * 2) <= self.currentSize:
            mc = self.minChild(i)
            if self.heapList[i] > self.heapList[mc]:
                tmp = self.heapList[i]
                self.heapList[i] = self.heapList[mc]
                self.heapList[mc] = tmp
            i = mc
    def minChild(self,i):
        if i * 2 + 1 > self.currentSize:
            return i * 2
        else:
            if self.heapList[i*2] < self.heapList[i*2 + 1]:
                return i * 2
            else:
                return i * 2 + 1
    ## 从二叉堆中删除最小的元素
    def delMin(self):
        retval = self.heapList[1]
        self.heapList[1] = self.heapList[self.currentSize]
        self.currentSize = self.currentSize - 1
        self.heapList.pop()
        self.percDown(1)
        return retval
        
    ## 根据元素列表构建堆
    def builgHeap(self,alist):
        i = len(alist) // 2
        self.currentSize = len(alist)
        self.heapList = [0] + alist[:]
        while (i > 0):
            self.percDown(i)
            i = i - 1

6.二叉搜索树

6.1搜索树的实现

二叉搜索树依赖性质:小于父节点的键都在左子树中,大于父节点的键则都在右子树。

代码实现:

class BinarySearchTree:
    def __init__(self):
        self.root = None
        self.size = 0
    def length(self):
        return self.size
    def __len__(self):
        return self.size
    def __iter__(self):
        return self.root.__iter__()
    # 插入新节点
    def put(self,key,val):
        if self.root:
            self._put(key,val,self.root)
        else:
            self.root = TreeNode(key,val)
        self.size = self.size + 1

    def _put(self,key,val,currentNode):
        if key < currentNode.key:
            if currentNode.hasLeftChild():
                self._put(key,val,currentNode.leftChild)
            else:
                currentNode.leftChild = TreeNode(key,val,parent=currentNode)
        else:
            if currentNode.hasRightChild():
                self._put(key,val,currentNode.rightChild)
            else:
                currentNode.rightChild = TreeNode(key,val,parent=currentNode)
    def __setitem__(self, key, value):
        self._put(key,value)

    ## 查找键对应的值
    def get(self,key):
        if self.root:
            res = self._get(key,self.root)
            if res:
                return res.payload
            else:
                return None
        else:
            return None
    def _get(self,key,currentNode):
        if not currentNode:
            return None
        elif currentNode.key == key:
            return currentNode
        elif key < currentNode.key:
            return self._get(key,currentNode.leftChild)
        else:
            return self._get(key,currentNode.rightChild)
    def __getitem__(self, key):
        return self.get(key)

    # 检查树中是否有某个键
    def __contains__(self, key):
        if self._get(key,self.root):
            return True
        else:
            return False
    # 删除
    def delete(self,key):
        if self.size > 1:
            nodeToRemove = self._get(key,self.root)
            if nodeToRemove:
                self.remove(nodeToRemove)
                self.size = self.size - 1
            else:
                raise KeyError("Error,key not in tree")
        elif self.size == 1 and self.root.key == key:
            self.root = None
            self.size = self.size - 1
        else:
            raise KeyError("Error,key not in tree")
    def __delitem__(self, key):
        self.delete(key)

    """
        1. 待删除节点没有子节点
        2. 待删除节点只有一个子节点
        3. 待删除节点有两个子节点
    """
    # 寻找后继结点
    def findSuccessor(self):
        succ = None
        if self.hasRightChild():
            succ = self.rightChild.findMin()
        else:
            if self.parent:
                if self.isLeftChild():
                    succ = self.parent
                else:
                    self.parent.rightChild = None
                    succ = self.parent.findSuccessor()
                    self.parent.rightChild = self
        return succ

    def findMin(self):
        current = self
        while current.hasLeftChild():
            current = current.leftChild
        return current

    def spliceOut(self):
        if self.isLeaf():
            if self.isLeftChild():
                self.parent.leftChild = None
            else:
                self.parent.rightChild = None
        elif self.hasAnyChildren():
            if self.hasLeftChild():
                if self.isLeftChild():
                    self.parent.leftChild = self.leftChild
                else:
                    self.parent.rightChild = self.leftChild
                self.leftChild.parent = self.parent

            else:
                if self.isLeftChild():
                    self.parent.leftChild = self.rightChild
                else:
                    self.parent.rightChild = self.rightChild
                self.rightChild.parent = self.parent

    def remove(self,currentNode):
        if currentNode.isLeaf():
            if currentNode == currentNode.parent.leftChild:
                currentNode.parent.leftChild = None
            else:
                currentNode.parent.rightChild = None
        elif currentNode.hasBothChildren():
            succ = currentNode.findSuccessor()
            succ.spliceOut()
            currentNode.key = succ.key
            currentNode.payload = succ.payload
        else:
            if currentNode.hasLeftChild():
                if currentNode.isLeftChild():
                    currentNode.leftChild.parent = currentNode.parent
                    currentNode.parent.leftChild = currentNode.leftChild
                elif currentNode.isRightChild():
                    currentNode.leftChild.parent = currentNode.parent
                    currentNode.parent.rightChild = currentNode.leftChild
                else:
                    currentNode.replaceNodeData(currentNode.leftChild.key,
                                                currentNode.leftChild.payload,
                                                currentNode.leftChild.leftChild,
                                                currentNode.leftChild.rightChild
                                                )
            else:
                if currentNode.isLeftChild():
                    currentNode.rightChild.parent = currentNode.parent
                    currentNode.parent.leftChild = currentNode.rightChild
                elif currentNode.isRightChild():
                    currentNode.rightChild.parent = currentNode.parent
                    currentNode.parent.rightChild = currentNode.rightChild
                else:
                    currentNode.replaceNodeData(currentNode.rightChild.key,
                    currentNode.rightChild.payload,
                    currentNode.rightChild.leftChild,
                    currentNode.rightChild.rightChild                            
                                                )
    # 二叉搜索树迭代器
    def __iter__(self):
        if self:
            if self.hasLeftChild():
                for elem in self.leftChild:
                    yield elem
            yield self.key
            if self.hasRightChild():
                for elem in self.rightChild:
                    yield elem

class TreeNode:
    def __init__(self,key,val,left = None,right = None,parent = None):
        self.key = key
        self.payload = val
        self.leftChild = left
        self.rightChild = right
        self.parent = parent

    def hasLeftChild(self):
        return self.leftChild
    def hasRightChild(self):
        return self.rightChild
    def isLeftChild(self):
        return self.parent and self.parent.leftChild == self
    def isRightChild(self):
        return self.parent and self.parent.rightChild == self
    def isRoot(self):
        return not self.parent
    def isLeaf(self):
        return not (self.rightChild or self.leftChild)
    def hasAnyChildren(self):
        return self.rightChild or self.leftChild
    def replaceNodeData(self,key,value,lc,rc):
        self.key = key
        self.payload = value
        self.leftChild = lc
        self.rightChild = rc
        if self.hasLeftChild():
            self.leftChild.parent = self
        if self.hasRightChild():
            self.rightChild.parent = self

7.平衡二叉搜索树(AVL树)

实现AVL树时,要记录每个节点的平衡因子。

平衡因子 = 左右子树的高度之差

→ 保证树的平衡因子为-1,0,1,可以使得关键操作获得更好的大O性能

#from 第6章树.二叉搜索树 import TreeNode
def _put(self, key, val, currentNode):
    if key < currentNode.key:
        if currentNode.hasLeftchi1d():
            self._put(key, val, currentNode.leftChild)
        else:
            currentNode.leftChild = TreeNode(key, val,parent=currentNode)
            self.updateBalance(currentNode.leftChild)
    else:
        if currentNode.hasRightChild():
            self._put(key, val, currentNode.rightChild)
        else:
            currentNode.rightchild - TreeNode(key, val,parent=currentNode)
            self.updateBalance(currentNode.rightChild)
def updateBalance(self, node):
    if node.balanceFactor > 1 or node.balanceFactor < -1:
        self.rebalance(node)
        return
    if node.parent != None:
        if node.isLeftChild():
            node.parent.balanceFactor += 1
        elif node.isRightChild():
            node.parent.balanceFactor -= 1
        if node.parent.balanceFactor != 0:
            self.updateBalance(node.parent)
# 实现左旋
def rotateLeft (self, rotRoot) :
    newRoot = rotRoot .rightchild
    rotRoot .rightChild = newRoot.leftChild
    if newRoot . leftChild !=None :
        newRoot . leftChild. parent = rotRoot
    newRoot.parent =rotRoot.parent
    if rotRoot .isRoot( ):
        self.root = newRoot
    else:
        if rotRoot .isLeftChild():
            rotRoot.parent .leftChild = newRoot
        else:
            rotRoot.parent .rightChild = newRoot
    newRoot . leftChild = rotRoot
    rotRoot.parent = newRoot
    rotRoot. balanceFactor = rotRoot . balanceFactor + 1 - min(newRoot . balanceFactor,0)
    newRoot . balanceFactor = newRoot . balanceFactor + 1 +max(rotRoot . balanceFactor,o )

# 实现再平衡
def rebalance(self, node) :
    if node. balanceFactor < 0:
        if node .rightChild .balanceFactor > 0:
            self.rotateRight (node.rightChild)self.rotateLeft (node)
        else:
            self.rotateLeft (node)
    elif node. balanceFactor > 0 :
        if node . leftChild. balanceFactor < 0:
            self.rotateLeft (node. leftChild)
            self.rotateRight (node)
        else:
            self.rotateRight (node)
nceFactor + 1 - min(newRoot . balanceFactor,0)
    newRoot . balanceFactor = newRoot . balanceFactor + 1 +max(rotRoot . balanceFactor,o )
# 实现再平衡
def rebalance(self, node) :
    if node. balanceFactor < 0:
        if node .rightChild .balanceFactor > 0:
            self.rotateRight (node.rightChild)self.rotateLeft (node)
        else:
            self.rotateLeft (node)
    elif node. balanceFactor > 0 :
        if node . leftChild. balanceFactor < 0:
            self.rotateLeft (node. leftChild)
            self.rotateRight (node)
        else:
            self.rotateRight (node)

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